In the first part of the talk, I will provide an introduction to finite element
methods for partial differential equations with input data affected by
uncertainty, described by random variables and random fields. Specifically, I
will present the basic theoretical and algorithmic aspects of the
Stochastic Galerkin and Stochastic Collocation finite element methods
for an elliptic model problem.
In the second part, I will present some results on my work on SPDE constrained optimization.
We consider an optimal control problem constrained by an elliptic SPDE, with a stochastic cost
functional of tracking type. We use a sparse grid stochastic collocation approach to discretize
in the probability space and finite elements to discretize in the physical space.
To accelerate the solution process, we propose a deterministic multigrid preconditioner for the
stochastic reduced KKT system, similar to the preconditioners introduced by
Draganescu and Dupont for the deterministic PDE constrained problem.

The first part of the talk covers difficulties associated with the dynamics of systems that lack the C1 smoothness (only piece-wise smooth); it includes the “anomaly” of having parts of the basins of attracting fixed points aggregating to into globally unbounded orbits.
The second part of the talk covers some analytical results regarding the bifurcations of a piecewise smooth dynamical system modeling cardiac arrhythmia. Earlier work (with Baker, Kidwell, and Kline) showed that as the duration of the basic cycle length (T) decreases, the dynamics gets richer. Generally speaking, the smaller the T, the more periodic orbits; moreover, large T correspond to global attractors, mid-range T have co-existing stable periodic orbits, and systems with small T have many unstable periodic orbits. These results, and experimental preparations, suggested that the bifurcation from stability to instability (as T decreases) is unique. The second part of the talk will disprove that, and also describe some subsets of the parameter region associated with unique bifurcations.

The interplay between fluid flow and diffusion of a solute in the fluid is
a primary mechanism for transport and mixing of substances, and one of the
most ubiquitous phenomena in nature. Since the seminal investigation by
G.I. Taylor in 1953, it has also been the focus of much mathematical efforts to model it.
Taylor's counterintuitive result -- that at long times the effective diffusivity
determined by the flow scales like the inverse of the tracer's molecular
diffusivity -- is a classical illustration of the power of mathematical analysis
and arguably one of the most remarkable effects demonstrate in this context.
This talk will report results that focus on the interaction of advection and diffusion
with fluid boundaries, such as pipes or ducts, at early and intermediate time
scales in the transport process. This can have direct applications to microfluidics.
Many microfluidic systems—including chemical reaction, sample analysis, separation,
chemotaxis, and drug development and injection—require control and precision of solute
transport. Although concentration levels are easily specified at injection, pressure-
driven transport through channels is known to spread the initial distribution, resulting
in reduced concentrations downstream. By monitoring the skewness (centered,
normalized third moment) of the tracer distribution in laminar, pressure driven flows an
unexpected phenomenon can be revealed: The channel’s cross-sectional aspect ratio
alone can control the shape of the concentration profile along the channel length.
Thin channels (aspect ratio << 1) deliver solutes arriving with sharp fronts and tapering tails,
whereas thick channels (aspect ratio ~ 1) produce the opposite effect. This occurs for
rectangular and elliptical pipes, independent of initial distributions. Thus, it is possible
to deliver solute with prescribed distributions, ranging from gradual buildup to sudden
delivery, based only on the channel dimensions.

We describe some mathematical models of how individuals interact
by influencing and being influenced by each other. In most generality,
such models include swarming behavior of birds and insects, herding behavior
of land animals as well as human social interactions. When the identity of
individuals known as "agents" is maintained and the number of agents is
finite, these models usually are ODEs or more generally SDEs or Markov
processes. In the continuum limit and with loss of identity of agents, these
models take the form of evolution equations for measures or PDEs.
We focus particularly on a model of opinion dynamics known as the
Hegselmann-Krause model and its generalizations. Equilibria of such models
consist of "clusters" where each cluster corresponds to a subset of agents
who have the same opinion. We present some recent results
on the analysis of a class of such ODE models. Our results include
Lyapunov stability as well as a form of robustness. The notion of robustness
studied is concerned with what happens to a system with finite
number of agents in equilibrium when a new agent is introduced. In particular,
we are concerned with whether the "disruption" to the system changes
continuously with the weight of the new agent.
This is joint work with Dr. Serap Tay Stamoulas.

The combination of powerful embedded computers, advanced sensor technology, and high speed wireless networks could revolutionize how we interact with our physical environment. Sensor networks that provide real time feedback offer significant value in terms of energy reduction, fault detection, equipment diagnostics, monitoring, security and more. This revolution will not happen in a positive way without a clear vision of how sense, network, and control technologies can be applied to enhance human abilities and improve our lives.
Such systems have been frustratingly difficult to implement. An old dilemma is becoming increasingly apparent. Networking provides remote access to information and control inputs. Gathering useful information, however, may require the installation of an expensive and intrusive array of sensors. Without this array, networked control provides colorful but minimally useful real information. Technological marvels like solid-state or micro-electromechanical sensors may ultimately reduce the cost of individual sensors through mass-production. They may not, however, reduce installation expense. They also do nothing to recover waste of resources. Even with the array, it may be difficult for a facilities operator to make informed control and maintenance decisions that intelligently affect mission critical components. Large datasets remain difficult to use.
This talk presents a design approach for creating cyber physical infrastructure that addresses these challenges to delivering actionable real time feedback. At the core of the system is a suite of non-intrusive sensors that dramatically reduce the cost of data acquisition. These sensors process and store data locally, without any dependency on external servers. This removes the security and privacy concerns that plague conventional sensor networks. Demonstrations will include a non-intrusive power meter that can measure multi-phase current and voltage without any contact to the power line, and a Wattsworth installation which reduces the fuel consumption of a US Army Forward Operating Base (FOB) by over 10 percent.

Mar

23

Multiscale modeling and simulation: some challenges and new perspectives

The design and optimization of the next generation of materials and applications strongly hinge on our understanding of the processing-microstructure-performance relations; and these, in turn, result from the collective behavior of materials’ features at multiple length and time scales. Although the modeling and simulation techniques are now well-developed at each individual scale (quantum, atomistic, mesoscale and continuum), there remain long-recognized grand challenges that limit the quantitative and predictive capability of multiscale modeling and simulation tools. In this talk we will discuss three of these challenges and provide solution strategies in the context of specific applications. These comprise (i) the homogenization of the mechanical response of materials in the absence of a complete separation of length and/or time scales, for the simulation of the dispersive nature of heterogeneous media; (ii) the collective behavior of materials’ defects, for the understanding of the kinematics of large elastoplastic deformations; and (iii) the upscaling of non-equilibrium material behavior for the modeling of phase change materials.

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge--Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing methods up seventh order accuracy. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.

Abstract: I will present an overview of a research program pursued by Irina Popovici and myself over several years developing a theory connecting equations of analytic curves in digital images with moment values of those images. Most proofs employ merely multivariable calculus and matrix theory. The theory has applications in digital image enhancement, image coding, and computer vision. I will also present some avenues inviting further exploration.

Feb

03

Finite and Infinite Population Spatial Replicator Dynamics from a Cellular Automaton Model

We derive finite and infinite population spatial replicator dynamics as the fluid limit of a cellular automaton. The infinite population spatial replicator is identical to the model used by Vickers and justifies the addition of a diffusion on the replicator. The finite population form fully generalizes the work of Durrett and Levin to arbitrary games and contains an intriguing nonlinear term that projects the species gradient onto the population gradient, resulting in behavior distinct from the infinite population case. We illustrate our findings with the rock-paper-scissors game and also show that the cellular automaton model can be used to understand qualitative solutions of the non-linear partial differential equations, especially with complex boundary conditions or non-differentiable starting conditions.

Around the world, sea levels are rising in response to warming oceans, melting glaciers, and shrinking ice sheets – and even faster rise is projected in the coming century. In this talk, Prof. Robert Kopp will explore the different physical processes driving sea-level rise, the geological record of past sea-level changes, methods for assessing the probability of different levels of future changes, and the implications for future coastal flood risks.